We show that the sheaf of ${\mathbb{A}}^1$-connected components of a reductive algebraic group over a perfect field is strongly ${\mathbb{A}}^1$-invariant. As a consequence, torsors under such groups give rise to ${\mathbb{A}}^1$-fiber sequences. We also show that sections of ${\mathbb{A}}^1$-connected components of anisotropic, semisimple, simply connected algebraic groups over an arbitrary field agree with their $R$-equivalence classes, thereby removing the perfectness assumption in the previously known results about the characterization of isotropy in terms of affine homotopy invariance of Nisnevich locally trivial torsors.

中文翻译：

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强 A1${\mathbb {A}}^1$-A1${\mathbb {A}}^1$-归约代数群连通分量的不变性

我们证明了${\mathbb{A}}^{\mathrm{1个}}$- 一个完美域上的还原代数群的连通分量是强的${\mathbb{A}}^{\mathrm{1个}}$-不变的。因此，这些组下的 torsors 会产生${\mathbb{A}}^{\mathrm{1个}}$-纤维序列。我们还表明部分${\mathbb{A}}^{\mathrm{1个}}$- 各向异性、半单、单连通代数群在任意域上的连通分量与它们一致$R$- 等价类，从而消除了先前已知结果中关于根据 Nisnevich 局部平凡 torsors 的仿射同伦不变性表征各向同性的完美性假设。